The electric field cavity array effect of 2D nano-sieves

For the upsurge of high breakdown strength (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{{{{\rm{E}}}}}}}_{{{{{{\rm{b}}}}}}}$$\end{document}Eb), efficiency (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{{{\rm{\eta }}}}}}$$\end{document}η), and discharge energy density (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{{{{\rm{U}}}}}}}_{{{{{{\rm{e}}}}}}}$$\end{document}Ue) of next-generation dielectrics, nanocomposites are the most promising candidates. However, the skillful regulation and application of nano-dielectrics have not been realized so far, because the mechanism of enhanced properties is still not explicitly apprehended. Here, we show that the electric field cavity array in the outer interface of nanosieve-substrate could modulate the potential distribution array and promote the flow of free charges to the hole, which works together with the intrinsic defect traps of active Co3O4 surface to trap and absorb high-energy carriers. The electric field and potential array could be regulated by the size and distribution of mesoporous in 2-dimensional nano-sieves. The poly(vinylidene fluoride-co-hexafluoropropylene)-based nanocomposites film exhibits an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{{{{\rm{E}}}}}}}_{{{{{{\rm{b}}}}}}}$$\end{document}Eb of 803 MV m−1 with up to 80% enhancement, accompanied by high \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{{{{\rm{U}}}}}}}_{{{{{{\rm{e}}}}}}}$$\end{document}Ue = 41.6 J cm−3 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{{{\rm{\eta }}}}}}\,$$\end{document}η≈ 90%, outperforming the state-of-art nano-dielectrics. These findings enable deeper construction of nano-dielectrics and provide a different way to illustrate the intricate modification mechanism from macro to micro.


Supplementary
Where E is the measure electric field at breakdown failure, E b is the characteristic breakdown strength corresponding to an ≈63% probability failure, and β ′ is a shape parameter to evaluate the data scatter.

Supplementary Note 5. Density functional theory (DFT) calculations for the interface region in external strong electric field
All DFT calculations are performed with the Vienna ab initio package (VASP). 14-16 Spin polarization was applied in all the calculations within the framework of DFT, using the PBE exchange-correlation functional, 17 with the dispersion interaction corrected by the D3 scheme. 18 The cutoff energy for the plane waves is 400 eV, and the atomic core region is described by PAW pseudopotentials. 19 The DFT + U method using the Dudarev approach 20 was employed for the transition metal oxides. The U here is the effective Hubbard Ueff =U-J, where J is equal to zero and U is 3.0eV for the Co. 21,22 For the surface calculation, dipole correction and potential correction along the z-direction are also taken into consideration. 23 Where E total , E surface , and E m are the energies of the system with molecule adsorption on the surface, the clean surface and the molecule, respectively. For charge density difference alone z axis calculation, charge density difference ρ ads is get first which is defined as ρ ads = ρ total − ρ surface − ρ m Where ρ total , ρ surface , and ρ m are the charge on the system with molecule adsorption on the surface, the clean surface and the molecule, respectively. Then, we get charge density difference alone z axis. For charge transfer calculation 27 , the Bader population analysis is used to get charge on the system.
Supplementary Figure 22. The calculated charge density difference between polymer and Co3O4.

Supplementary Note 6. Macro-finite element simulation (FES) revised by DFT results to analyze the mechanism of nano-fillers
At a given operating field, the electric field (E) and electric potential (φ) in the P(VDF-HFP)-based nanocomposites films can be expressed as following specialized Maxwell's equations.
Where ε 0 is vacuum dielectric constant (≈8.854187817×10 -12 F/m), K is relative dielectric constant, E is electric field, ρ is space charge density, J is current density, φ is electric potential, σ is electrical conductivity. E distortion is highly detrimental to capacitors if its degree is overlarge because it degrades not only the dielectric performance, but also the device reliability and service lifetime. In this section, based on the result of DFT that the intermolecular charge transfer in Co3O4-P(VDF-HFP) inhibits the polariton of interfacial polymer molecules at high E, we revise the interface region parameters in model and simulate the steady-state internal E and φ distribution in nanocomposites by using finite element computations. Electric field modeling is complicated for nano-modified polymer matrix composites because there is significant variability in materials' properties, which are determined by temperature, orientation, crystallinity, etc. Note that the focus of this work is development of high electric field strength resistant insulation materials. The simulation involved in this study is simplified with several assumptions being made: 1) The samples are placed in an electric field of 200MV/m to explore the partial E distortion and potential distribution around the single nanoparticles, nanosheets, or nano-sieves. 2) 2D nanofillers are placed horizontally perpendicular to the direction of the applied electric field. 3) the polymer substrate is supposed to be homogeneous everywhere, with no distinction between crystalline and amorphous regions. 4) the parameters of nanofillers' properties are constant overtime in each case such that the partial electrical field and potential of dielectrics are only a function of morphology of nanofillers. 5) The K and σ values of components in nanocomposites stays unchanged in the simulation. The details of the simulation are discussed in the rest part of this section. The radius of nanoparticles is 8 nm. In order to make the obtained images present better visual effects, we set the length and width of the 2D nanomaterials to 200×500 nm. The specific model figures are shown in Supplementary Figure 21 In contrast to the routine COMSOL electric field simulation of nanoparticles-modified dielectrics that only consider two parts, fillers and substrates, does not in-depth consider about the interface region, which is generally recognized to be the part that plays a key role in enhancing the electrical performance of the dielectrics. Thus, a shell layer 0.5 nm thick is coated on the surface of the nanofillers to represent the interfacial region where the polarization of the polymer molecule is affected according to the results of first-principles calculations in the section 5. The K of upper interface layer (UIL) is set to 14, the K of lower interface layer (LIL) is set to 10. This is a rough qualitative correction. As shown in Supplementary Fig. 25a and b, it is well-observed that the electric potential lines are more intensive on the upper and lower sides of nanoparticles because of the mis-match in K values of the fillers and polymer matrix, which results in the concentrated electric field on the filler-polymer interfaces. In the simulation result ( Supplementary Fig. 25c) without interfacial layer, the E distortion on the upper and lower surfaces caused by nanoparticles is symmetric. But, in the simulation plot ( Supplementary Fig. 25d) with interfacial layer correction, the E distortion on the upper and lower surfaces caused by nanoparticles is asymmetric. The enhanced polymer molecular polarization of UIL achieves to a better buffer matching in fillerpolymer, which results in a reduced degree of E distortion. This might be essential to achieve increased breakdown strength of dielectric nanocomposites. Therefore, the breakdown mechanism of nano-modified dielectrics could be explicated more reasonably by using interfacial layer correction method. As shown in Fig. S26, from the column on the left, there are some points can be obtained. NPs/P(VDF-HFP) generate a downward conical potential trap with a minimum value obtained at the position of the center of the circle corresponding to the nanoparticle. Nanosheet has a concave basin-like potential trap. When 5nm pores appeared on the nanosheet, some upward protrusions began to appear at the bottom of the concave basin. As the pore size increases to 15nm, the bumps grow upwards, forming what resembles a tubular array. When the pore size reaches the maximum value of 50 nm, the protrusions grow further and are almost equal to the theoretical potential. When the pore diameter reaches a maximum value of 50 nm, the protrusions grow further almost equal to V theory , but the size limitation makes their quantities decrease sharply. The E distribution diagram on the right column also shows a similar pattern. Figure 29. 3D graphs of V r of (a) nanoparticle, (c) nanosheet, (e) 5@NSIs, (g) 15@NSIs, and (i) 50@NSIs. 3D graphs of normE of (b) nanoparticle, (d) nanosheet, (f) 5@NSIs, (h) 15@NSIs, and (j) 50@NSIs.

Supplementary
From the column on the left, there are some points can be obtained. NPs/P(VDF-HFP) generate a downward conical potential trap with a minimum value obtained at the position of the center of the circle corresponding to the nanoparticle. Nanosheet has a concave basin-like potential trap. When 5nm pores appeared on the nanosheet, some upward protrusions began to appear at the bottom of the concave basin. As the pore size increases to 15nm, the bumps grow upwards, forming what resembles a tubular array. When the pore size reaches the maximum value of 50 nm, the protrusions grow further and are almost equal to the theoretical potential. When the pore diameter reaches a maximum value of 50 nm, the protrusions grow further almost equal to V theory , but the size limitation makes their quantities decrease sharply. The E distribution diagram on the right column also shows a similar pattern. Figure 30. The planar potential distribution 3D graphs of V r of 15@NSIs near the interface (a) 100nm, (b) 50 nm, (c) 5nm, (d) 1 nm, and (e) 0.5 nm. In the region far from the interface (>20 nm), the planar potential distribution exhibits a relatively weaker and wider pit trap. Its effect is similar to that of nanosheets.

Supplementary
However, in the region of 0~10 nm, the planar potential shows a intricate 3D trap array distribution, reflecting the electric field cavity effect caused by the porous structure of the nano-sieve. The Distance from Interface (nm)